Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Tetali, Prasad; Committee Member: Duke, Richard; Committee Member: Heitsch, Christine; Committee Member: Randall, Dana; Committee Member: Trotter, William T.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 115-119).

The main results of this dissertation are various algorithms related to partially ordered sets. The dissertation basically consists of two parts. The first part treats algorithms that generate ideals of partially ordered sets. The second part concerns the generation of partially ordered sets themselves. First, we present two algorithms for listing ideals of a forest poset. These algorithms generate ideals in a Gray Code manner, that is, consecutive ideals differ by exactly one element. Both algorithms use storage O(n), where n is the number of elements in the poset. The first algorithm traverses, at each phase, the current ideal being listed and runs in time O(nN), where N is the number of ideals of the poset. The second algorithm mimics the first but eliminates the traversal and runs in time O(N). This algorithm has the property that the amount of computation between successive ideals is O(1). Secondly, we give orderly algorithms for constructing acyclic digraphs, acyclic transitive digraphs, finite topologies and finite topologies and finite lattices. For the first time we show that the number of finite lattices on 11, 12, and 13 elements are 37622, 262775, and 2018442, respectively, and the number of finite topologies on 8 and 9 elements are 35979 and 363083, respectively. We also describe orderly algorithms for generating k-colored graphs. We present, in particular, an algorithm for generating connected bicolorable graphs. We also prove some properties of a canonic matrix which might be generally useful for improving the efficiency of orderly algorithms. / Graduate

The purpose of this paper is to present a study of partly ordered sets. It includes a rigorous development of relations based on the notion of a relation as a set, lattices, and theorems concerning the lattice of subgroups of a group.

This thesis is concerned with the development of two generalized set covering models. The first model is formulated for the total set covering problem where cost is minimized subject to the constraint that each customer must be served by at least one facility. The second model is constructed for the partial set covering problem in which customer coverage is maximized subject to a budget constraint. The conventional formulations of both the total set covering and partial set covering problems are shown to be special cases of the two generalized models that arc developed. Appropriate solution strategies arc discussed for each generalized model. A specialized algorithm for a particular case of the partial covering problem is constructed and computational results are presented. / M.S.

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.

Thesis (Ph.D.)--Syracuse University, 2008. / "Publication number: AAT 3323082."

Modern software systems have been pervasively concurrent to utilize parallel hardware and perform asynchronous tasks. The correctness of concurrent programming, however, has been challenging for real-world and large systems. As the concurrent events of a system can interleave arbitrarily, unexpected interleavings may lead the system to undefined states, resulting in denials of services, performance degradation, inconsistent data, security issues, etc. To detect such concurrency errors, concurrency testing repeatedly explores the interleavings of a system to find the ones that induce errors. Traditional systematic testing, however, suffers from the intractable number of interleavings due to the complexity in real-world systems. Moreover, each iteration in systematic testing adjusts the explored interleaving with a minimal change that swaps the ordering of two events. Such exploration may waste time in large homogeneous sub-spaces leading to the same testing result. Thus on real-world systems, systematic testing often performs poorly to reveal even simple errors within a limited time budget. On the other hand, randomized testing samples interleavings of the system to quickly surface simple errors with substantial chances, but it may as well explore equivalent interleavings that do not affect the testing results. Such redundancies weaken the probabilistic guarantees and performance of randomized testing to find any errors. Towards effective concurrency testing, this thesis leverages partial order semantics with randomized testing to find errors with strong probabilistic guarantees. First, we propose partial order sampling (POS), a new randomized testing framework to sample interleavings of a concurrent program with a novel partial order method. It effectively and simultaneously explores the orderings of all events of the program, and has high probabilities to manifest any errors of unexpected interleavings. We formally proved that our approach has exponentially better probabilistic guarantees to sample any partial orders of the program than state-of-the-art approaches. Our evaluation over 32 known concurrency errors in public benchmarks shows that our framework performed 2.6 times better than state-of-the-art approaches to find the errors. Secondly, we describe Morpheus, a new practical concurrency testing tool to apply POS to high-level distributed systems in Erlang. Morpheus leverages dynamic analysis to identify and predict critical events to reorder during testing, and significantly improves the exploration effectiveness of POS. We performed a case study to apply Morpheus on four popular distributed systems in Erlang, including Mnesia, the database system in standard Erlang distribution, and RabbitMQ, the message broker service. Morpheus found 11 previously unknown errors leading to unexpected crashes, deadlocks, and inconsistent states, demonstrating the effectiveness and practicalness of our approaches.

No description available.